If you’re going to include all of the negative numbers, you might as well include (a, 0) and (0, b) as well. As a variant you could plot only primitive triples (including if you like (±1, 0) and (0, ±1)).

I had the same thought on that second comment, and made this https://observablehq.com/@jeffbaumes/relatively-prime-pythagorean-triples

P.S. to understand why the parabolas show up, see https://www.youtube.com/watch?v=QJYmyhnaaek